Romance
Romance is, naturally, one of the biggest genres of a story centered around hormonal teenagers. It is divided roughly into two categories, het (as in "heterosexual") and slash (homosexual). Some may argue this generalization, pointing to the Aberforth/goat and Fred/mirror stories that floated around Fiction Alley for awhile, but these are so scarce that they don't really count as a subgenre.
Het is the more popular of the two kinds of romance. Popular ships, or pairings, are Harry/Ginny, Ron/Hermione, Lily/James, and Harry/Hermione. Almost every fic site will have het, unless it is a specialized slash archive.
Slash is subdivided into slash and femmeslash. Slash is male/male, and tends to be a great deal darker than het. Popular pairings include Harry/Draco, Remus/Sirius, and Snape/Harry (Snarry). Femmmeslash, due to the lack of female characters in the series, is not quite as popular, but thanks to sites such as Raspberry Swirl and movements like the Femmeslash Revolution on Fiction Alley, it is gaining momentum. F/F is usually very dark and you cannot find all that many chaptered fics. Popular ships are Hermione/Ginny, Hermione/Cho, and Ginny/Luna.

AngstAngst is what it sounds like: dark, depressing, generally monologues. This can be found in almost any fic you read, other than fluffy romances. Slash seems particularly prone to angst. Frequently angsty!characters include: Snape, Harry, Remus, Draco, and Sirius. Really disturbing, dark angst is dark!fic or darkfic.

Action/Adventure/General
These fics generally follow the structure and tone of the books. Many of them were written in the so-called "three-year summer" between books 4 and 5, giving us a plethora of 5th year fics to read through. This genre is prone to Mary Sues at times.

Mary Sues (and Gary Stus)
Ah, the dreaded Mary Sue. In the Harry Potter fandom, Mary Sue is used not only to describe when an author inserts herself into the story, but a perfect, wonderful, absolutely, sickeningly flawless and flat original character or a real character changed into a perfect person. These tend to be beautiful, female, American exchange students hiding dark secrets who fall in love with either Harry, Draco, Snape, Remus, or Sirius (perhaps not coincidentally the series’ favorite male characters). A male Mary Sue (Gary Stu), appears far less, usually in slash fic. These can be found in all fandoms, but due to the relative young age of some writers in this one, they seem to be rather pervasive. Generally, sites such as fanfiction.net will have rather a lot of them, but quality control sites like Fiction Alley will not.

Small List of Terms

Puppyshipper: Person who likes Remus/Sirius.
Snarry: Snape/Harry
AU: Alternate Universe
cookie: preview for a longer fic
drabble: A very short fic. Fiction Alley gives the definition as one less than or exactly 100 words, but that can be stretched.
founderfic: a fiction about the founders Gryffindor, Ravenclaw, Hufflepuff, and Slytherin.
MWPP: Stands for Moony, Wormtail, Padfoot, and Prongs, the four marauders. Set pre-1981, and while they were in school.
Squick: a squick is something that grosses you out.
Twincest: Fred/George or Padma/Pavarti

Famous and/or Good Fics

You'd have to be blind, deaf, dumb, and otherwise incapacitated to be in the fandom more than a week and not hear about Cassandra Claire's Draco trilogy. This is perhaps the most famous of all fanfics, many claiming it is better than the actual books. Cassie Claire is also famous in the LOTR community as the author of the Very Secret Diaries. I recommend reading this fic, because you will be considered an ignorant n00b if you don't.
Nemesis's I Am Lord Voldemort is also very famous. It is a fabulous story chronicling Tom Riddle's descent into darkness.Dreamwalk Blue by Viola is also great, but unfinished. It is centered around Albus Dumbledore, Tom Riddle, and a host of engaging, believable original characters.
Iniga's Darkness Dying is another classic, answering the eternal (to some, anyways...) question, Why did Sirius suspect Remus? Truly well written, though somewhat out of date after OOtP.

Fiction Archives

http://www.fictionalley.org FictionAlley
The biggest, most comprehensive fanfiction site out there. It is divided into the Dark Arts for angst, Schnoogle for novel-length fics, the Astronomy Tower for romance, Riddikulus for humor, and Fiction Alley Park forum. There is also Artistic Alley for fanart and Lumos Dissendium for writing tips. Most of the big fish in the fandom can be seen here every once and a while. The forums are newbie friendly, which is great, and there are many wonderful services to help you improve your writing. All fics screened for quality-control!

http://www.sugarquill.net The Sugar Quill
This is a wonderful community that is very strict about canon (what JKR has actually written and is likely to write). Therefore, Ron/Hermione, Ginny/Harry, and like ships are encouraged and the Weasleys are doted upon. Though not exactly slash-hostile, it's not a big place for it. However, Harry/Hermione shippers are generally politely, if coldly, turned down. Nevertheless, the Sugar Quill is a nicely selective, but warm group of writers, just make sure to read their rules.

http://www.werewolfregistry.net The Werewolf Registry
This is a wonderful Remus Lupin-centric domain. Lots of slash and angst, but there is het too. If you like Lupin, I heartily reccommend it.

So...

I hope that this small primer on the wonderful world of Harry Potter fanfiction has perked your interest. New authors are always welcome and reviewers are adored, but be aware: this is a huge fandom and you won't be famous in it if you're mediocre. That being said, it is usually friendly and warm. So, have fun slashing, shipping, sailing, reccing and r/r-ing in this diverse, varied, and entertaining fandom!